GMS Equations From Irreversible Thermodynamics ChEn 6603 References • E. N. Lightfoot, Transport Phenomena and Living Systems, McGraw-Hill, New York 1978. • R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena 2 nd ed., Chapter 24 McGraw-Hill, New York 2007. • D. Jou, J. Casas-Vazquez, Extended Irreversible Thermodynamics, Springer-Verlag, Berlin 1996. • R. Taylor, R. Krishna Multicomponent Mass Transfer, John Wiley & Sons, 1993. • R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, London, 1969. 1 Saturday, March 26, 2011
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GMS Equations From Irreversible
ThermodynamicsChEn 6603
References• E. N. Lightfoot, Transport Phenomena and Living Systems, McGraw-Hill, New York 1978.
• R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena 2nd ed., Chapter 24 McGraw-Hill, New York 2007.
• D. Jou, J. Casas-Vazquez, Extended Irreversible Thermodynamics, Springer-Verlag, Berlin 1996.
• R. Taylor, R. Krishna Multicomponent Mass Transfer, John Wiley & Sons, 1993.
• R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, London, 1969.
1Saturday, March 26, 2011
Outline
Entropy, Entropy transportEntropy production: “forces” & “fluxes”• Species diffusive fluxes & the Generalized Maxwell-Stefan Equations
• Heat flux
• Thermodynamic nonidealities & the “Thermodynamic Factor”
Example: the ultracentrifugeFick’s law (the full version)Review
2Saturday, March 26, 2011
A PerspectiveReference velocities• Allows us to separate a species
flux into convective and diffusive components.
Governing equations• Describe conservation of mass,
momentum, energy at the continuum scale.
GMS equations• Provide a general relationship
between species diffusion fluxes and diffusion driving force(s).
• So far, we’ve assumed:‣ Ideal mixtures (inelastic collisions)
‣ “small” pressure gradients
Goal: obtain a more general form of the GMS equations that represents more physics• Body forces acting differently
on different species (e.g. electromagnetic fields)
• Nonideal mixtures
• Large pressure gradients (centrifugal separations)
3Saturday, March 26, 2011
EntropyEntropy differential:
Total (substantial/material) derivative:Specific volumev
ρ = 1/v
Dρ
Dt= −ρ∇ · v ρ
Dωi
Dt= −∇ · ji + σi
DDt≡ ∂
∂t+ v ·∇
Chemical potentialper unit massµi = µi/MiTds = de + pdv −
n
i=1
µidωi
Internal energye
TρDs
Dt= ρ
De
Dt+ pρ
Dv
Dt−
n
i=1
µiρDωi
Dt
TρDs
Dt= ρ
De
Dt− p
ρ
Dρ
Dt−
n
i=1
µiρDωi
Dt
ρDe
Dt= −∇ · q− τ : ∇v − p∇ · v +
n
i=1
fi · ji
4Saturday, March 26, 2011
Entropy Transport
chain rule...
∇(αβ) = α∇β + β∇α
TρDs
Dt= −∇ · q− τ : ∇v − p∇ · v +
n
i=1
fi · ji +p
ρρ∇ · v −
n
i=1
µi (−∇ · ji + σi) ,
= −∇ · q− τ : ∇v +n
i=1
fi · ji +n
i=1
µi∇ · ji −n
i=1
µiσi,
ρDs
Dt= −∇ ·
1T
q−
n
i=1
µiji
Transport of s
+q ·∇
1T
−
n
i=1
ji ·∇
µi
T
− 1
Tτ : ∇v +
1T
n
i=1
fi · ji −1T
n
i=1
µiσi
Production of s
5Saturday, March 26, 2011
ρDs
Dt= −∇ · js + σsNow let’s write this in the form:
Look at this term(entropy production due to species diffusion)
ρDs
Dt= −∇ ·
1T
q−
n
i=1
µiji
Transport of s
+q ·∇
1T
−
n
i=1
ji ·∇
µi
T
− 1
Tτ : ∇v +
1T
n
i=1
fi · ji −1T
n
i=1
µiσi
Production of s
js =1T
q−
n
i=1
µiji
σs = q ·∇
1T
−
n
i=1
ji ·∇
µi
T
− 1
Tτ : ∇v +
1T
n
i=1
fi · ji −1T
n
i=1
µiσi,
= −qT
·∇ lnT −n
i=1
ji ·∇
µi
T
− 1
Tfi
− 1
Tτ : ∇v − 1
T
n
i=1
µiσi
Tσs = −q ·∇ lnT −n
i=1
ji ·∇T,pµi +
Vi
Mi∇p− fi
Λi
−τ : ∇v −n
i=1
µiσi
diffusive transport of entropy
production of entropy
∇
µi
T
=
∂µi
∂T∇
T
T
+
1T
∂µi
∂p∇p +
1T∇T,pµi,
=1T
1
Mi
∂µi
∂p∇p +∇T,pµi
,
=1T
Vi
Mi∇p +∇T,pµi
6Saturday, March 26, 2011
Part of the Entropy Source Term…Why can we add this “arbitrary” term?What does this term represent?
From physical reasoning (recall di represents force per unit volume driving diffusion) or the Gibbs-Duhem equation,
n
i=1
di = 0
ωi
Mi=
xi
M
φi = ciVi
µi =µi
Mi
ji = ρωi (ui − v)
Vi Partial molar volume.
cRTdi = ci∇T,pµi + (φi − ωi)∇p− ωiρ
fi −
n
k=1
ωkfk
n
i=1
ji ·∇T,pµi +
Vi
Mi∇p− fi
Λi
=n
i=1
ji ·
Λi −1ρ∇p +
n
k=1
ωkfk
n
i=1
ji · Λi =n
i=1
ρωi(ui − v) ·
∇T,pµi +
Vi
Mi− 1
ρ
∇p− fi +
n
k=1
ωkfk
,
=n
i=1
(ui − v) ·
ci∇T,pµi + (φi − ωi)∇p− ρωi
fi −
n
k=1
ωkfk
cRTdi
,
= cRTn
i=1
di · (ui − v),
= cRTn
i=1
1ρωi
di · ji
7Saturday, March 26, 2011
The Entropy Source Term - Summary
Interpretation of each term???
ρDs
Dt= −∇ · js + σs
From the previous slide:n
i=1
ji · Λi = cRTn
i=1
di · jiρi
js =1T
q−
n
i=1
µiji
cRTdi = ci∇T,pµi + (φi − ωi)∇p− ωiρ
fi −
n
k=1
ωkfk
Tσs = −q ·∇ lnT −n
i=1
ji ·∇T,pµi +
Vi
Mi∇p− fi
Λi
−τ : ∇v −n
i=1
µiσi
= −q ·∇ lnT 1
−n
i=1
cRT
ρidi · ji
2
− τ : ∇v 3
−n
i=1
µiσi
4
8Saturday, March 26, 2011
σs ∼ Forces ⋅ Fluxes
Fundamental principle of irreversible
thermodynamics:
σs =
α
JαFα
Tσs = −q ·∇ lnT −n
i=1
cRT
ρidi · ji − τ : ∇v −
n
i=1
µiσi
Flux, Jα Force, Fα
q −∇ lnTji − cRT
ρidi
τ −∇v
Lαβ - Onsager (phenomenological) coefficients
Jα = Jα(F1, F2, . . . , Fβ ; T, p, ωi)
Lαβ ≡∂Jα
∂Fβ
Jα =
β
∂Jα
∂Fβ
Fβ +O (FβFγ)
≈
β
LαβFβ
Fluxes are functions of:• Thermodynamic state variables:
T, p, ωi.
• Forces of same tensorial order (Curie’s postulate)‣What does this mean?
‣More soon…
Lαβ = Lβα
9Saturday, March 26, 2011
Species Diffusive FluxesTensorial order of “1” ⇒ any vector force may contribute.
From irreversible thermo:
Fick’s Law:
Generalized Maxwell-Stefan Equations:
Index form: n-1 dimensional matrix form
DTi - Thermal Diffusivity
Flux: Jα q ji τForce: Fα −∇ lnT − cRT
ρidi −∇v
ρ(d) = −[Bon](j)−∇ lnT [Υ](DT )di = −n
j =i
xixj
ρÐij
jiωi− jj
ωj
−∇ lnT
n
j =i
xixjαTij
αTij =
1Ðij
DT
i
ρi− DT
i
ρj
Dij - Fickian diffusivity
ji = −n
j=1
LijcRT
ρjdj − Liq∇ lnT
ji = −ρn
j=1
Dijdj −DTi ∇ lnT
(j) = −ρ [L] (d) +∇ lnT (βq)
(j) = −ρ[D](d)− (DT )∇ lnT
10Saturday, March 26, 2011
Constitutive Law: Heat Flux
Choose Lqq=λT to obtain “Fourier’s Law”
“Dufuor” effect - mass driving force can cause heat flux!
Usually neglected.
Tensorial order of “1” ⇒ any vector force may contribute.
Flux: Jα q ji τForce: Fα −∇ lnT − cRT
ρidi −∇v
q = −Lqq∇ lnT −n
i=1
LqicRT
ρidi
The “Species” term is typically included here, even though it does not come from irreversible thermodynamics. Occasionally radiative terms are also included here...
here we have substituted the RHS of the GMS
equations for di.q = −λ∇T
Fourier
+n
i=1
hiji Species
+n
i=1
n
j =i
cRTDT xixj
ρiÐij
jiρi− jj
ρj
Dufour Note: the Dufour effect
is usually neglected.
11Saturday, March 26, 2011
Observations on the GMS Equations
What have we gained?• Thermal diffusion (Soret/Dufuor)
& its origins.‣ Typically neglected.
• “Full” diffusion driving force‣Chemical potential gradient (rather than
mole fraction). More later.‣ Pressure driving force.‣ When will φi ≠ ωi? More later.
‣ Body force term.‣ Does gravity enter here?
Onsager coefficients themselves not too important from a “practical” point of view.Still don’t know how to get the binary diffusivities.